Syllabus

W. Griffiths. Introduction to Quantum Mechanics. (Required)
C. Cohen-Tannoudji. Quantum Mechanics. Vol. 2. (Required)
J. J. Sakurai. Modern Quantum Mechanics. (Recommended if you like it; somewhat advanced)
R. Shankar. Principles of Quantum Mechanics. (Recommended if you like it; somewhat advanced)

You must complete 8.05 - "Quantum Physics II", with a grade of C or better before attempting 8.06.

Grades will be determined by a weighted average of problem sets (30%), a Midterm (15%), a Term Paper (20%), and a Final Exam (35%). The faculty may alter grades to reflect class participation, improvement, effort and other qualitative measures of performance.

Problem sets are a very important part of 8.06. We believe that sitting down yourself and trying to reason your way through a problem not only helps you learn the material deeply, but also develops analytical tools fundamental to a successful career in science. We recognize that students also learn a great deal from talking to and working with each other. We therefore encourage each 8.06 student to make his/her own attempt on every problem and then, having done so, to discuss the problems with one another and collaborate on understanding them more fully. The solutions you submit must reflect your own work. They must not be transcriptions or reproductions of other people's work. Plagiarism is a serious offense and is easy to recognize. Don't submit work which is not your own. We do not accept problem sets after they are due. Period. However, your lowest problem set score will be discarded at the end of the semester; only the remaining n - 1 will be used in determining your grade.

Everyone in 8.06 will be expected to research, write and "publish" a short paper on a topic related to the content of 8.05 or 8.06. The paper can explain a physical effect or further explicate ideas or problems covered in the courses. It can be based on the student's own calculations and/or library research. The paper should be written in the and format of a brief journal article and should aim at an audience of 8.06 students. Writing, editing, revising and "publishing" skills are an integral part of the project, which is described in full in a separate handout.

I begin this outline with an abbreviated outline for 8.05, Quantum Physics II, to remind you of what we have already done.

   8.05

1. General structure of quantum mechanics.
   
2. Quantum dynamics.
   
3. Two-state systems.
   
4. Angular momentum in quantum mechanics and the hydrogen atom, neglecting spin.
   
5. Spin.
   
6. Addition of angular momentum.
   
7. Introduction to the quantum mechanics of identical particles and to degenerate Fermi systems.
   

   8.06

1. Natural Units.
   
   Reading: Supplementary notes.
   
   • The cgs system of units.
   • Natural units.
   • Examples.
     
     
2. Charged particles in a magnetic field.
   
   Reading: Supplementary notes. Griffiths, Section 10.2.4; Cohen-Tannoudji, Ch. VI Complement E.
   • Canonical quantization.
   • The classical Lagrangian and Hamiltonian for a particle in a static magnetic field.
   • The quantum mechanical analysis of a charged particle in a magnetic field, via canonical quantization.
   • Landau levels. Energy eigenvalues. Energy eigenstates. Energy eigenvalues in another gauge.
   • Gauge invariance and the Schrödinger equation.
   • Landau level wave functions. Counting the states in a Landau level.
   • de Haas–van Alphen effect.
   • Integer Quantum Hall Effect. Introduction to the ordinary Hall effect. Quantum mechanical problem of a particle in crossed magnetic and electric fields. Calculation of Hall current due to a single filled Landau level. From this idealized calculation to real systems: the role of impurities.
   • The Aharonov-Bohm effect.
     
     
3. Time-independent perturbation theory.
   
   Reading: Griffiths, Ch. 6; Cohen-Tannoudji, Ch. XI including Complements A-D; Cohen-Tannoudji, Ch. XII. If you wish, see also Shankar, Ch. 17 and Sakurai, Ch. 5.1-3.
   • Time-independent perturbation theory for degenerate states: diagonalizing perturbations and lifting degeneracies.
   • Time-independent perturbation theory for nondegenerate states: Energy and wavefunction perturbations through second order.
   • Degeneracy reconsidered.
   • Simple examples: perturbing a two-state system, a simple harmonic oscillator, and a bead on a ring.
   • The fine structure of hydrogen, revisited: relativistic and spin-orbital effects.
   • The hydrogen atom in a magnetic field, revisited: the Zeeman effect.
   • The hydrogen atom in a electric field: the Stark effect.
   • Van der Waals interaction between neutral atoms.
     
     
4. Variational and semi-classical methods.
   
   Reading: Griffiths, Chs. 7, 8; Cohen-Tannoudji, Ch. XI Complements E, F, G. If you wish, see also Shankar, Ch. 16 and Sakurai, Ch. 5.4.
   • The variational method.
   • Ground state of helium. Screening.
   • First excited state of helium. Direct and exchange integrals.
   • A one electron molecule (H₂⁺ ).
   • The Semi-classical (or WKB) approximation. Form of wave functions in classically allowed and classically forbidden regions. Handling turning points: connection formulae. Tunnelling. Semiclassical approximation to bound state energies.
     
     
5. Quantum Computing.
   • Using many two-state systems as a quantum computer.
   • Grover algorithm. Shor algorithm.
     
     
6. The adiabatic approximation and Berry’s phase.
   
   Reading: Griffiths, Ch. 10.
   
   • The Born-Oppenheimer approximation and the rotation and vibration of molecules.
   • The adiabatic theorem.
   • Application to spin in a time-varying magnetic field.
   • Berry’s phase, and the Aharonov-Bohm effect revisited.
   • Resonant adiabatic transitions and The Mikheyev-Smirnov-Wolfenstein solution to the solar neutrino problem.
     
     
     
7. Scattering.
   
   Reading: Griffiths, Ch. 11; Cohen-Tannoudji, Ch. VIII. If you wish, see also Shankar, Ch. 19.
   • Definition of cross-section σ and differential cross section dσ/dΩ. General form of scattering solutions to Schrödinger equation, the definition of scattering amplitude f, and the relation of f to dσ/dΩ. Optical theorem.
   • The Born approximation. Derivation of Born approximation to f. Application to scattering from several spherically symmetric potentials, including Yukawa and Coulomb. Scattering from a charge distribution.
   • Low energy scattering. The method of partial waves. Definition of phase shifts. Relation of scattering amplitude and cross section to phase shifts. Calculation of phase shifts. Behavior at low energies. Scattering length. Bound states at threshold. Ramsauer-Townsend effect. Resonances.
     
     
8. Time-dependent perturbation theory.
   
   Reading: Griffiths, Ch. 9; Cohen-Tannoudji, Ch. XIII. If you wish, see also Shankar, Ch. 18 and Sakurai, Ch. 5.5-8.
   
   • General expression for transition probability. Adiabatic theorem revisited.
   • Sinusoidal perturbations. Transition rate.
   • Emission and absorption of light. Transition rate due to incoherent light. Fermi’s Golden Rule.
   • Spontaneous emission. Einstein’s A and B coefficients. How excited states of atoms decay.